Probit Model

The probit model is a binary-outcome model that estimates $\Pr(y = 1 \mid \mathbf{x}) = \Phi(\mathbf{x}'\boldsymbol\beta)$, where $\Phi$ is the standard-normal CDF. By squashing the linear prediction $\mathbf{x}'\boldsymbol\beta \in (-\infty, +\infty)$ into $[0, 1]$, probit cures the LPM's out-of-range-probability problem; the price is non-linearity in parameters, so the model is estimated by MLE rather than OLS.

When to use

Use probit when the LPM's unbounded predictions or constant marginal effects are uncomfortable — typically when many fitted values approach 0 or 1, or when the substantive question is about probabilities at the extremes. In practice probit and logit give nearly identical predictions; the choice rarely matters. Raw coefficients $\hat\beta_j$ tell you the direction only; for the probability change you need the marginal effect $\varphi(\bar{\mathbf{x}}'\hat{\boldsymbol\beta}) \cdot \hat\beta_j$ (see the Computing marginal effects (logit / probit) recipe). Hypothesis testing uses the Likelihood Ratio Test rather than the F-test.